Geodesic
Handling ray transform of symmetric tensor fields and the Radon transform of differential forms on incomplete data
Daghestan Electronic Mathematical Reports, no. 1 (2014), pp. 56-70 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

A symmetric tensor field in the Euclidian space of any dimension ${{\mathbb R}}^n$ can be reconstructed from its integrals along straight lines which form n-manifold. We obtained the new formulas reconstructing solenoidal part of the fields when a ray transform is known only on complex lines, intersecting a given curve, as well as a curve belonging to infinity. The new family of two-dimensional planes in ${{\mathbb R}}^n$ is introduced and the inversion formula is obtained for the case of differential forms of degree $2$ on the given integrals along the planes of this family.
Keywords: symmetric tensor field, ray transform, complex of lines, Radon transform of differential forms.
Mots-clés : reconstruction, solenoidal part 
@article{DEMR_2014_1_a1,
     author = {Z. G. Medzhidov},
     title = {Handling ray transform of symmetric tensor fields and the {Radon} transform of differential forms on incomplete data},
     journal = {Daghestan Electronic Mathematical Reports},
     pages = {56--70},
     year = {2014},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DEMR_2014_1_a1/}
}
TY  - JOUR
AU  - Z. G. Medzhidov
TI  - Handling ray transform of symmetric tensor fields and the Radon transform of differential forms on incomplete data
JO  - Daghestan Electronic Mathematical Reports
PY  - 2014
SP  - 56
EP  - 70
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/DEMR_2014_1_a1/
LA  - ru
ID  - DEMR_2014_1_a1
ER  - 
%0 Journal Article
%A Z. G. Medzhidov
%T Handling ray transform of symmetric tensor fields and the Radon transform of differential forms on incomplete data
%J Daghestan Electronic Mathematical Reports
%D 2014
%P 56-70
%N 1
%U http://geodesic.mathdoc.fr/item/DEMR_2014_1_a1/
%G ru
%F DEMR_2014_1_a1
Z. G. Medzhidov. Handling ray transform of symmetric tensor fields and the Radon transform of differential forms on incomplete data. Daghestan Electronic Mathematical Reports, no. 1 (2014), pp. 56-70. http://geodesic.mathdoc.fr/item/DEMR_2014_1_a1/

[1] Denisjuk A., “Inversion of the X-ray transform for 3D symmetric tensor fields with sources on a curve”, Inverse problems, 22 (2006), 399–411 | DOI | MR | Zbl

[2] Palamodov V.P., “Reconstruction of a differential form from Doppler transform”, SIAM J. Math. Anal., 41 (2009), 1713–1720 | DOI | MR | Zbl

[3] Sharafutdinov V.A., “Slice-by-slice reconstruction algorithm for vector tomography with incomplete data”, Inverse problems, 23 (2007), 2603–2627 | DOI | MR | Zbl

[4] Vertgeim L.B., “Integral geometry problems for symmetric tensor fields with incomplete data”, J. Inverse III-Posed Probl., 8(3) (2000), 355–364 | MR | Zbl

[5] Sharafutdinov V.A., Integral Geometry of Tensor Fields, VSP, Utrech, 1994 | MR

[6] Grangeat P., “Mathematical framework of cone-beam 3D-reconstruction via the first derivative of the Radon transform”, Lecture Notes in Math., 1497, 1990, 66–97 | DOI | MR

[7] Natterer F., Matematicheskie aspekty kompyuternoi tomografii, Mir, M., 1990 | MR

[8] Medzhidov Z.G., “Vosstanovlenie vektornogo polya po dannym ego luchevogo preobrazovaniya na trekhmernykh mnogoobraziyakh pryamykh”, Vestnik DGU, 2012, no. 6, 159–166